↑

Properties

Label 14.0.31626503554...2299.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,19^{7}\cdot 29^{12}$
Root discriminant $78.14$
Ramified primes $19, 29$
Class number $4544$ (GRH)
Class group $[4, 4, 284]$ (GRH)
Galois group $C_{14}$ (as 14T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1493501, -625188, 959807, -343260, 279108, -86598, 44797, -11303, 4381, -863, 288, -49, 20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 20*x^12 - 49*x^11 + 288*x^10 - 863*x^9 + 4381*x^8 - 11303*x^7 + 44797*x^6 - 86598*x^5 + 279108*x^4 - 343260*x^3 + 959807*x^2 - 625188*x + 1493501)
 
gp: K = bnfinit(x^14 - 5*x^13 + 20*x^12 - 49*x^11 + 288*x^10 - 863*x^9 + 4381*x^8 - 11303*x^7 + 44797*x^6 - 86598*x^5 + 279108*x^4 - 343260*x^3 + 959807*x^2 - 625188*x + 1493501, 1)
 

Normalized defining polynomial

\( x^{14} - 5 x^{13} + 20 x^{12} - 49 x^{11} + 288 x^{10} - 863 x^{9} + 4381 x^{8} - 11303 x^{7} + 44797 x^{6} - 86598 x^{5} + 279108 x^{4} - 343260 x^{3} + 959807 x^{2} - 625188 x + 1493501 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-316265035547780605989332299=-\,19^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.14$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $19, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(551=19\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{551}(1,·)$, $\chi_{551}(227,·)$, $\chi_{551}(132,·)$, $\chi_{551}(455,·)$, $\chi_{551}(170,·)$, $\chi_{551}(400,·)$, $\chi_{551}(210,·)$, $\chi_{551}(20,·)$, $\chi_{551}(286,·)$, $\chi_{551}(343,·)$, $\chi_{551}(248,·)$, $\chi_{551}(436,·)$, $\chi_{551}(284,·)$, $\chi_{551}(94,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} - \frac{4}{17} a^{9} - \frac{4}{17} a^{8} - \frac{4}{17} a^{6} + \frac{5}{17} a^{5} - \frac{6}{17} a^{4} + \frac{4}{17} a^{3} + \frac{7}{17} a^{2} + \frac{1}{17} a$, $\frac{1}{17} a^{11} - \frac{3}{17} a^{9} + \frac{1}{17} a^{8} - \frac{4}{17} a^{7} + \frac{6}{17} a^{6} - \frac{3}{17} a^{5} - \frac{3}{17} a^{4} + \frac{6}{17} a^{3} - \frac{5}{17} a^{2} + \frac{4}{17} a$, $\frac{1}{17} a^{12} + \frac{6}{17} a^{9} + \frac{1}{17} a^{8} + \frac{6}{17} a^{7} + \frac{2}{17} a^{6} - \frac{5}{17} a^{5} + \frac{5}{17} a^{4} + \frac{7}{17} a^{3} + \frac{8}{17} a^{2} + \frac{3}{17} a$, $\frac{1}{32903291628400118271412547423} a^{13} - \frac{807784048513719894165318142}{32903291628400118271412547423} a^{12} - \frac{188735863466790396954411690}{32903291628400118271412547423} a^{11} - \frac{226468723159612779145778402}{32903291628400118271412547423} a^{10} + \frac{253409941364700427393530199}{1935487742847065780671326319} a^{9} + \frac{16447357766034119915885774648}{32903291628400118271412547423} a^{8} + \frac{2834188163561683030188031608}{32903291628400118271412547423} a^{7} - \frac{28746141854533405642578660}{32903291628400118271412547423} a^{6} + \frac{5769063003892400402246102431}{32903291628400118271412547423} a^{5} - \frac{13122790837545666846549701069}{32903291628400118271412547423} a^{4} + \frac{740289038602904010367900246}{32903291628400118271412547423} a^{3} + \frac{11790727036762957832847944673}{32903291628400118271412547423} a^{2} + \frac{3408877683676717715149810752}{32903291628400118271412547423} a + \frac{500799871958200860111105672}{1935487742847065780671326319}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{4}\times C_{4}\times C_{284}$, which has order $4544$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $6$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6020.985100147561 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-19}) \), 7.7.594823321.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ R ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$19$19.14.7.2$x^{14} - 376367048 x^{2} + 3575486956$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$